Analytical Gradient Evaluation of Cost Functions in. General Field Solvers: A Novel Approach for. Optimization of Microwave Structures

Transcription

1 IMS 2 Workshop Analytcal Gradent Evaluaton of Cost Functons n General Feld Solvers: A Novel Approach for Optmzaton of Mcrowave Structures P. Harscher, S. Amar* and R. Vahldeck and J. Bornemann* Swss Federal Insttute of Technology, IFH ETH Zentrum, Glorastrasse 35, CH Zürch *Department of Electrcal and Computer Engneerng, Unversty of Vctora BOX 355, Vctora, B.C. V8W 3P6

2 IMS 2 Workshop 2 Abstract Ths paper ntroduces a method for the analytcal calculaton of gradents of a cost functons whch s an attractve feature when optmzng mcrowave structures usng feld solvers. In contrast to utlzng fnte dfferencng all gradents are computed from a sngle analyss of the structure regardless of ts complexty. It s not even necessary to nvert a large matrx; a lnear system [A][x]=[b] s solved nstead. No remeshng s requred n the FEM and the gradent values are exact. The basc technque used n ths new approach s appled to a moment method technque (CIET) and the fnte element method (FEM). Both methods lend tself to the approprate matrx equaton. Introducton Evaluatng the gradent of a cost functon for the optmzaton of mcrowave crcuts s usually based on the fnte dfference technque and can be a tme consumng task. Ths s especally true when the crcut transfer functon s calculated on the bass of a feld-theory smulaton tool snce always two computatons are necessary for one gradent. If, n addton, the number of ndependent varables s large, optmzaton can become an mpossble task. In ths contrbuton t wll be shown that, under certan crcumstances, the gradent of a cost functon can be calculated analytcally wthout usng fnte dfferences. The number of computatons can be cut n half and well known dsadvantages wth fnte dfferencng lke naccuraces at sngulartes n hghly resonant structures are elmnated. The method has been appled successfully to the coupled ntegral equaton technque (CIET) and the fnte element method (FEM). Analytcal calculaton of the cost functon s also possble wth the adjont network method (ANM) but requres a network representaton of the structure to be optmzed (and ts adjont). The mode matchng technque (MMT) to calculate the felds s normally utlsed to extract the network s representaton n form of

3 IMS 2 Workshop 3 the admttance or scatterng matrx. The ANM has been successfully appled to the optmzaton of flters and radatng structures. Analytcally calculatng the gradent of a cost functon drectly n general numercal technques wthout frst dervng a network representaton has not been publshed before. The possblty of dong so s of great nterest as t offers a number of obvous advantages and also not so obvous ones, dependng on the numercal method used. The new approach can not be appled to all numercal feld computaton methods snce t requres a scatterng problem representaton of the whole mcro-wave structure of the form whch results drectly from applyng the FEM or a moment method (.e. CIET), but s not necessarly lmted to these methods. Here, [A] s a M x M matrx whch depends on the ndependent varables and represents the structure to be optmzed, [b] s the exctaton and [x] s the response. For example, the vector [x] contans the expanson coeffcents n the MoM or the nodal values n FEM. It wll be shown, that as long as the partal dervatves of the matrx [A] and the exctaton [b] are known analytcally, all senstvtes can be determned analytcally. Up to now ths approach has been successfully tested wth the MoM n the optmzaton of wavegude flters and was subsequently appled to the FEM. It can be extended to other methods for whch the scatterng problem can be formulated as above. The advantages of ths approach are summarzed as follows: No network representaton needed; only one cost functon evaluaton nstead of two; hgher accuracy compared to a fnte dfference scheme n partcular n the vcnty of resonances; no remeshng of the structure requred durng gradent calculatons; no matrx nverson necessary; reduced memory requrements; faster algorthms.

4 IMS 2 Workshop 4 OUTLINE Introducton Revew Analyss of Mcrowave Structures Optmzaton Analytc Gradent Evaluaton Example CIET Example FEM Concluson

5 IMS 2 Workshop 5 INTRODUCTION Desgn of Mcrowave Devces and Structures conssts of: Modellng Analyss Optmzaton Optmzaton can be done by: Stochastc Methods Determnstc Methods (.e. Gradent Methods) Goal: Mnmze a Cost Functon of the form: : constants Ê 6 6 n n n n opt Fa ( ) = K S ω, a - S ω, a 2 K a n ω n : parameters : frequences

6 IMS 2 Workshop 6 REVIEW The problem s the calculaton of the gradents What has been done n the past? a Fa ( ) n general: calculaton of the gradents wth fnte dfference optmzaton of mcrowave flters and radatng structures by the Adjont Network Method (ANM) ½ network representaton of the mcrowave crcut (MMT) ½ analytcal evaluaton of network senstvtes (gradents) NEW: Analytcal calculaton of gradents (due to geometrc changes) of cost functons when usng more general technques lke Fnte Element Method (FEM) or Method of Moments (MoM).

7 IMS 2 Workshop 7 numercal methods lke: ANALYSIS OF MICROWAVE STRUCTURES Fnte Element Method (FEM) Coupled Integral Equaton Technque (CIET) Spectral Doman Approach (SDA) Frequency Doman Transmsson Lne Matrx Method (FDTLM) Method of Moments (MoM) Method of Lnes (MoL) :? :? = Å Matrx equaton Z k S

8 IMS 2 Workshop 8 OPTIMIZATION We use: determnstc, gradent based method mnmze a cost functon of the form: Ê 6 6 n n n n opt Fa ( ) = K S ω, a - S ω, a 2 wth frst dervatves: = Ê - a Fa K S a S a S a opt n, ( ) ω n ω n, ω 9 n a n a : varables,.e. geometry parameters 6

9 IMS 2 Workshop 9 Characterzaton of the problem: multdmensonal nonlnear functon of several varables constrant varables Sutable methods: Gauss Newton Levenberg Marquardt use of frst dervatves (Numercal recpes, NAG lbrary, Matlab optmzaton toolbox)

10 IMS 2 Workshop ANALYTIC GRADIENT EVALUATION = Ê - a Fa K S a S a S a opt n, ( ) ω n ω n, ω 9 n a :? :? Matrx equaton: Z k = S n 6 6 ;@ S ω, a = const. * fct( a )* k n j S a = Re % & ' S S S a ( ) *

11 IMS 2 Workshop frst possblty: = a S a a k ω n,... 6 ;@ :? :? Z k = S ;@ ;@ ;@ + = a Z k Z a k a S j a ;@ - k = Z ;@ S - a a Z;@ k, wth matrx nverson Z - second possblty: SOLVE lnear system for a k wthout nvertng matrx Z :

12 IMS 2 Workshop 2 FIRST EXAMPLE: CIET I II III IV XIV L L2 L3 L3 XV Output E-Plane-Stubflter b b b2 b3 b3 X X2 X3 X4 X3 X4 z= Equatons for r E and r H n all regons example:! 5 " $#! 5 y I - jk I z π jk I nz Ê b y n n= Š I E = e + B cos n- e x I - jk I z π jk I nz b y Ê n n n= Š I I I H =- Y e + Y B cos n- e " $# y z 7 stubs Longtudnal Secton Electrc (LSE)-Modes k Y m m π m = k - a - - π 2 b 2 π k - a = ωµ k m 2 5 b 2 2 Input TE

13 IMS 2 Workshop 3 :? :? The result s a matrx equaton of the form Z k = S! A B a C D E b K F G H c ¼& AJ AK AL # K m AM AN n " $ % ' ( K ) K * = % K & K ' U ( K ) K * unknown coeffcents: ap, bp... np, p= M M: number of bass functons U p : exctaton and the matrx elements Ap, Bp,... ANp, example Š Š I bb Ap j Yn B B bb = Ê p + ÊB bb B U p = n= I jy B bb n n n bb II p n kn L b cot n= b b b

14 IMS 2 Workshop 4 Bass functons: B bjbl J n 5 b 6 / π j π n = 5¼ b 5 5 l + 2 G n 2 3 π b - - n- 2 b! 5 5 Solve for the unknown coeffcents a, b,... j l b b j l 6 / " b j b l 6 / j $ # l J π - + n- 6 / π 2 5 5b - + n- b Wthn the CIET the reflecton coeffcent S s gven n terms of the spectrum of the bass functons and the expanson coeffcents a T bb bb S =- + B () a =- + a B () :? M Ê =

15 IMS 2 Workshop 5 6 RESULTS sngle E-plane stub n a rectangular wavegude real part of S / L 3 magnary part of S / L sold lne: analytc dashed lne: fnte dfference δ a = a/ Real( S / L) 3 2 sold lne: analytc dashed lne: fnte dfference Imag( S / L) 2 3 lne: analytc sold lne: analytc dashed lne: fnte dfference δ a = a/ dashed lne: fnte dfference F (GHz) F (GHz) F (GHz) F (GHz) stub length L=a, fnte dfference ncrement L=a/

16 IMS 2 Workshop 6 6 RESULTS sngle E-plane stub n a rectangular wavegude real part of S / L magnary part of S / L sold lne: analytc dashed lne: fnte dfference δ a = a/ 2 Real( S / L) 3 2 dashed sold lne: analytc lne: fnte dfference Imag( S / L) 2 3 analytc sold lne: analytc dashed lne: fnte dfference δ a = a/ dashed lne: fnte dfference F (GHz) F (GHz) F (GHz) F (GHz) stub length L=a, fnte dfference ncrement L=a/

17 IMS 2 Workshop 7 RESULTS 7 E-plane stubs n a rectangular wavegude " / H K A = O B. % I J K > / I! " # $ % & '! C I τ & $ " ) = O JE?. E EBBA HA? A & " $ & ". / F (GHz) group delay

18 IMS 2 Workshop 8 RESULTS Bandpass Flter Center frequency: GHz Band wdth: % Mnmum return loss n passband: 26 db

19 IMS 2 Workshop 9 SECOND EXAMPLE: FEM Parallel plate wavegude wth delectrc nsert A B 3.5 cm h y x = 5 cm x x 2 z Helmholtz equaton, 2-D, cartesan coordnate system x Hz + ε x y ε r r H y z 2 + k H = S = fct( H z ) at plane A µ r z + boundary condtons

20 IMS 2 Workshop 2 Use of a trangular mesh, lnear nterpolaton functons L e j e e e ( x, y) = ( a + b x + c y), j =,2, 3 2 e j j j a, e e e j, bj c j : constant coeffcents = functon of trangle coord. example: e e e e e a = x2 y3 y2 x3 e : area of a trangle ndces : node number e: element number The H Feld wthn trangle e s: H e z 3 ( ) e x, y = L j ( x, y) j= H e z, j

21 IMS 2 Workshop 2 Applyng Galerkn Method the result s a matrx equaton of the form: ;@ K Hz = b K : system matrx, wth elements comng from e K elemental values: elemental matrces K ;@ b : exctaton vector e e e e e e 2 D Kj = b b c c k e e j + j - r µ δ D ε r j Solve matrx equaton for unknown coeffcents H z S can be formulated n terms of H z S = H z ( x = ) H j at cuttng plane A: - Ã ne (,), n ( je,)

22 IMS 2 Workshop 22 ANALYTIC GRADIENT EVALUATION S = - jkx Hz ( x) - He Hz ( x = ) = jkx He H -, x s set to zero S h = Hz ( x = ) H h Å calculate dervatves H = z ( x ) h

23 IMS 2 Workshop 23 Hz ( x = ) h lnear set: K h H elements of z z h H z = h b - h K H nodes on cuttng plane A. ;@ = h b, because exctaton s ndependent of h. h K, dervatves of the elements of the system matrx: e e e e e e 2 D Kj = b b c c k e e j + j - r µ δ D ε r j h K e j =... h= h

24 IMS 2 Workshop 24 RESULTS h Mesh settng, trangular mesh wth 88 elements and 6 nodes e For calculaton of h= h only nodes along the red edge needed. h K j

25 IMS 2 Workshop h (cm) R T Analytcal FD h (cm) ( ), ( ) S analytcal h FD wth h =. cm. S h S h 2

26 IMS 2 Workshop 26 CONCLUSIONS novel technque to evaluate gradents for optmzaton general approach for problems that can be formulated n terms of general nonhomogeneous matrx equatons (e.g. FEM, MoM) gradents are determned analytcally (exact) no need for network representaton of the problem no fnte dfferencng only one functon evaluaton needed (only one mesh settng) no matrx nverson result: fast and accurate optmzaton of mcrowave structures excellent agreement between present approach and fnte dfference